The short answer is 'No'.
This saying, which I first heard from a Swedish friend some years ago, refers to the 'pet names' that parents and other family members give to their children. It does not mean 'alternative formal callings' but affectionate diminutives: nick-names, bynames.
Each of these aliases usually reflects some aspect of the child's personality which is endearing; some milestone in their development; some amusing or in other way remarkable episode. The namings are ways of expressing love for the child, by underlining their uniqueness.
Examples abound in my own family, and now with my own children, but I shall offer none. This is, by the way, known as 'Family Speak' and Nancy Keesing has written of and documented some of it.
I thought the 'Eskimos and snow' myth had been debunked. English has many names for precipitation from the sky: mist, drizzle, downpour, spitting, shower, deluge... These describe the nature of the rain, how it arrives from above. They do not suggest that it is anything other than rain. I understand that the Inuit 'snow' nomenclature is likewise.
'A rose by any other name' is not quite the same thing.
Romeo, Romeo. So how come you're all like 'Romeo' and stuff?
Juliet asks herself why she is so besotted with the name 'Romeo', and realises that she would be just as obsessed were the lad named Tyrell or Dennis. So to speak.